I wonder, if I have to analyze some function and find global extrema.

for example this function:

enter image description here

It has 2 equal local maxima. So what local maximum should I settle as global? Can I both or have to pick one? I don't know. Will you help me with this?

  • $\begingroup$ Do you mean what the maximum value of the function is, or where it occurs? $\endgroup$ – MPW Dec 26 '19 at 20:07
  • $\begingroup$ Since the global maximum is the value attained, not the point at which it is attained, this is a non-problem. $\endgroup$ – Bernard Dec 26 '19 at 20:07
  • $\begingroup$ I mean when I have to state global maximum of this function, should I say that there are 2 global maxima or only choose one and note that they are 2 equal local maxima? $\endgroup$ – naruto25 Dec 26 '19 at 20:12
  • 1
    $\begingroup$ It's fine to state that there are two global maximum points. $\endgroup$ – Ted Shifrin Dec 26 '19 at 20:14

A direct consequence of the defintion of global maxima is that the global maximum is necessarily unique, for if two real numbers $x,\ y$ are distinct, then by the order on $\mathbb{R}$ it must be the case that one is bigger than the other. What it can happen is that there are multiple, even an infinite amount of argmax, i.e. points whose image through a function gives the function's maximum. Therefore

$$f:\mathbb{R}\to\mathbb{R}:x\mapsto\sqrt{x^2-x^4}$$ $$\Rightarrow\max f= \frac{1}{2}, \text{argmax} f=\{-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\}$$


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